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**Lecture 12: Stochastic Discount Factor and GMM Estimation**

The following topics will be covered: SDF Basic expression Risk free rate Risk correction Mean-variance frontier Time-varying expected returns GMM GMM overview Applying GMM Also, more on “Hypothesis Testing” L12: SDF

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**Stochastic Discount Factor Presentation**

L12: SDF

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**Stochastic Discount Factor**

L12: SDF

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**Examples Asset price Stock return Excess stock return Risk free rate**

See page 9 – 10 of Cochrane. L12: SDF

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**Relating to EGS Recall the consumption and saving in Chapter 6, EGS:**

β is a discount factor for delaying consumption The first order condition of this problem is similar to the SDF presentation Also on page 176, EGS, we have Z is the aggregate wealth in state z, q(z) denotes the expected payoff of the firm conditional on z. In equilibrium, the marginal utility of the representative agent in a given state equals the equilibrium state price itself. Subject to wealth constraint on page 91 L12: SDF

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Risk-free rate L12: SDF

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Risk Corrections The first term is the present value of E(x) (expected payoff). The second is a risk adjustment. An asset whose payoff co-varies positively with the discount factor has its price raised and vice versa. The key u’(c) is inversely related to c. If you buy an asset whose payoff covaries negatively with consumption (hence u’(c)), it helps to smooth consumption and so is more valuable than its expected payoff indicates. L12: SDF

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**Risk Corrections – Return Expression**

All assets have an expected return equal to the risk-free rate, plus a risk adjustment. Assets whose returns covary positively with consumption make consumption more volatile, and so must promise higher expected returns to induce investor to hold them, and vice versa. L12: SDF

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**Expected Return-Beta Representation**

Where βis the regression coefficient of the asset return on m. It says each expcted return should be proportional to the regression coefficient in a regression of that return on the discount factor m. λis interpreted as the price of risk and β is the quantity of risk in each asset. L12: SDF

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**Mean-Variance Frontier**

Implications: Means and variances of asset returns lie within efficient frontier. On the efficient frontier, returns are perfectly correlated with the discount factor. The priced return is perfectly correlated with the discount factor and hence perfectly correlated with any frontier return. The residual generates no expected return. L12: SDF

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**Sharpe Ratio and Equity Premium Puzzle**

Let Rmv denote the return of a portfolio on the mean-variance efficient frontier and consider power utility. The slope of the frontier (Sharpe ratio) is Sharpe ratio is higher if consumption is more volatile or if investors are more risk averse. Over the last 50 years, average real stock return is 9% with a standard deviation of 16%. The real risk free rate is 1%. This suggests a real Sharpe ratio of _____ Aggregate nondurable and services consumption growth has a standard deviation of 1%. So L12: SDF

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**Time-varying Expected Returns**

The relation above is conditional. Conditional mean or other moment of a random variable could be different from its unconditional moment. E.g,, knowing tonight’s weather forecast, you can better predict rain tomorrow than just knowing the average rain for that date. It suggests a link between conditional mean of stock returns and conditional variance of stock returns. Little empirical support. L12: SDF

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Estimating SDF -- GMM L12: SDF

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**Estimating SDF – Second Stage**

L12: SDF

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Implementing GMM L12: SDF

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GMM Example L12: SDF

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GMM Example (2) L12: SDF

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GMM Example (3) L12: SDF

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**Program GMM using SAS /* N=5, 7 instruments */ proc model data=gmm;**

parms beta 1.0 gamma 1.0; endogenous cons0 cons1; exogenous r1 r2 r3 r4 r5; instruments lrm1 lrm2 lrm3 lrm4 lrm5 lrm6; eq.m1=1-(1+r0)*(beta*(cons0/cons1)**(-gamma); fit m1-m6/gmm kernel=(parzen, 1,0); ods output EstSummaryStats=parms; run; L12: SDF

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**More on Hypothesis Testing**

Testing J linear Restrictions We can base a test of H0 on the Wald criterion: The chi-squared statistic is not usable when σ2 is unknown. As an alternative, we have the following F statistic L12: SDF

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**Examples of J Restrictions**

Each row of R is a single linear restriction on the coefficient vector. L12: SDF

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More Examples L12: SDF

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**Example: Test of Structural Change**

L12: SDF

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**Test Based on Loss of Fit**

Least squares vector b is chosen to maximize R2. The overall fitness of a regression: To see if the coefficient of a particular variable is a given value, we can also apply the F-stat, where F[1,n-K]=t2[n-K] To see if the constraints on a set of variables hold, we use or L12: SDF

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Exercises CR 1.7 Read through Chapter 11, CR L12: SDF

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